CHARACTERIZING IGNEOUS ROCKS AND MINERALS USING MATRIX ANALYSIS

A matrix that describes an igneous rock is a rectangular array of numbers, each (row, column) matrix element representing moles of a cation (Si, Al, etc.) in a mineral or endmember. The matrix is an extensive variable (i.e., additive), but also intensive in the sense of specifying relative proportions of atoms (e.g., ab endmember = 3Si + Al + Na). Given a hand-sample bulk chemical analysis (in matrix form) and a mineral matrix postulated from petrographic observation, an algorithm in MATLAB code systematically varies the abundances of chemical constituents (e.g., Fein pyroxene) to reduce the calculated residual (mismatch) to zero: the search arrives at a perfect fit.

This methodology determines (i) what minerals/endmembers are present (or not), (ii) their relative molar abundances, and (iii) their chemical compositions, even of complex minerals such as amphibole. It provides an “instant” modal analysis characterizing all major and trace minerals of primary, xenocrystic, and alteration origin. If mis-calibration had biased the chemical analysis, the matrix procedure can even (iv) detect and correct for bad data. Calculated data are bulk properties on the scale of an analyzed hand sample, and not on a micro-scale such as compositional variations within a zoned crystal.

Complex processes alter a magma composition from inception of melting until eruption or emplacement. Phantom minerals are not present in the final product; they were either stranded at depth, or once existed but dissolved during ascent. Corroded clinopyroxene or epidote comprise “almost” phantom minerals. When applied to a suite of related samples, the matrix procedure can (v) identify a phantom mineral species and (vi) determine its composition. Examples: phantom olivine of fa:fo = 55:45 composition in Dokhan Volcanics, Egypt, and phantom clinopyroxene of hd:di = 39:61 in the São Rafael pluton, Brazil.